Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations

Authors:	Martin Hadac

Affiliation:	Mathematical Institute of the University of Bonn, Beringstraße 1, D-53115 Bonn, Germany

Abstract:	We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space $H^{s_1,s_2}(mathbb{R}^2)$ with $s_1>-frac12$ and . On the $H^{s_1,0}(mathbb{R}^2)$ scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for the following generalisation of the KP II equation: $displaystyle (u_t - vert D_xvert^alpha u_x + (u^2)_x)_x + u_{yy} = 0, quad u(0) = u_0,$ for $frac43<alphaleq 6$ , $s_1>max(1-frac34 alpha,frac14-frac38 alpha)$ , and $u_0in H^{s_1,s_2}(mathbb{R}^2)$ . We deduce global well-posedness for , and real valued initial data.

Keywords:	Kadomtsev-Petviashvili II equation Cauchy problem local well-posedness.

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